Conclusions

In the last few years, the hydrogeologic community has been exploring the use of geostatistics. As is to be expected in any new field, there have been examples of misapplication of methodologies and the reader often encounters in the literature conflicting statements and claims. There is no scarcity of sophisticated techniques but their use is not always tempered with a basic understanding of statistical methodology and the use of plain common sense. Widespread slips are using models with too many parameters, confusing fitting with prediction, confusing simulation with prediction, using an unnecessarily complex empirical model, neglecting to check whether the empirical model is consistent with the data, using a method outside of its intended scope, and adopting inconsistent assumptions.

Nevertheless, significant progress has been made and hydrogeologists now have a better grasp of the scope of geostatistics as well as of statistics in general than ever before. In fact, not only is geostatistical research vibrant and exciting but hydrogeology is leading the way on many fronts by proposing and testing innovative methodologies. Summarizing, some of the most important areas of current and needed research are:

• Improvement of linear geostatistics, which include adjustments of methods from analysis of variance, time series, regression, and Kalman filtering. The trend is to recognize gradually that many successful methods that appear under different names in different fields are basically the same. At the same time, it is hoped that geostatistics will remain pragmatic and focused on the analysis of geophysical data.

• Development of nonparametric and resampling estimation methods. Such methods could be quite useful because they would allow the analysis to proceed without hypotheses concerning distributions or specific values of statistical parameters. So far, such methods have been found useful in relatively simple problems but this is an important area of current research in statistics [see Lall [this issue]).

• Development of Bayesian methods. Usually, the error associated with parameter estimation is quite large. In some cases, the uncertainty in the parameters affects the predictions significantly.
  Bayesian methods can be used to account for the effect of parameter uncertainty and to incorporate information from other sources.

• Development of methods that utilize geological and other information. Important steps have been made but the emphasis so far has been on stochastic simulation and less on estimation, i.e., how to extract the information from the data. Problems of parameter estimation and model validation are much more challenging than in linear geostatistics.

• Development of nonlinear and nonGaussian estimators in the utilization of the groundwater flow and transport equations. The mathematical problems are quite challenging but estimation can be improved by considering the mechanics of flow and transport. The real challenge will be how to strike the right balance between practicality and exactitude.

• Combination of information from all sources of data, including geophysical and tracer data, principles of flow and tranport, and geological understanding.

• There is a need for amalgamation of geostatistical estimation methods with ``upscaling'' methods which recognize the differences between parameters or obervations that are defined at different scales.

Acknowledgments. Funding for this work was provided by the office of Research and Development, U.S. Environmental Protection Agency, under agreement R-815738-01 through the Western Region Hazardous Substance Research Center. The content of this study does not necessarily represent the views of the agency. I thank Keerthi Angammana for his help in conducting the bibliographic search. This review is not meant to be comprehensive but rather an evaluation of the overall progress and of remaining problems in the field.